Hardy-Lieb-Thirring inequalities for eigenvalues of Schrödinger operators

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Hardy-Lieb-Thirring inequalities for eigenvalues of Schrödinger operators

RUPERT L. FRANK

Doctoral Thesis Stockholm, Sweden 2007

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TRITA-MAT-07-MA-03 ISSN 1401-2278

ISRN KTH/MAT/DA 07/03-SE ISBN 978-91-7178-626-5

Kungl Tekniska högskolan Institutionen för matematik SE-100 44 Stockholm SWEDEN Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan fram- lägges till offentlig granskning för avläggande av filosofie doktorsexamen i matematik onsdagen den 9 maj 2007, klockan 9.00 i F3, KTH, Lindstedtsvä- gen 26, Stockholm.

° Rupert L. Frank, maj 2007c Tryck: Universitetsservice US-AB

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Abstract

This thesis is devoted to quantitative questions about the discrete spectrum of Schrödinger-type operators.

In Paper I we show that the Lieb-Thirring inequalities on moments of negative eigenvalues remain true, with possibly different constants, when the critical Hardy weight is subtracted from the Laplace operator.

In Paper II we prove that the one-dimensional analog of this inequality holds even for the critical value of the moment parameter.

In Paper III we establish Hardy-Lieb-Thirring inequalities for fractional powers of the Laplace operator and, in particular, relativistic Schrödinger operators. We do so by first establishing Hardy-Sobolev inequalities for such operators. We also allow for the inclusion of magnetic fields.

As an application, in Paper IV we give a proof of stability of relativistic matter with magnetic fields up to the critical value of the nuclear charge.

In Paper V we derive inequalities for moments of the real part and the modulus of the eigenvalues of Schrödinger operators with complex- valued potentials.

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Acknowledgements

First of all I would like to express my deep and sincere gratitude to my supervisor Ari Laptev. He introduced me to the topic of spectral estimates and shared many beautiful mathematical problems and ideas with me. His enthusiasm, encouragement and understanding have provided an excellent basis for the present thesis.

I am indebted to my teachers Mikhail Sh. Birman, Bernard Helffer, Elliott H. Lieb and Heinz Siedentop for all the time and patience they in- vested in my mathematical education. I would like to thank the Department of Mathematics at Université Paris Sud and the Department of Physics at Princeton University for their hospitality. Partial financial support through the ESF Scientific Programme in Spectral Theory and Partial Differential Equations (SPECT) and by the Swedish Foundation for International Co- operation in Research and Higher Education (STINT) is gratefully acknowl- edged.

I had the pleasure to collaborate with Tomas Ekholm, Pavel Exner, Anders M. Hansson, Ari Laptev, Elliott H. Lieb, Oleg Safronov, Robert Seiringer, Roman G. Shterenberg, Heinz Siedentop and Simone Warzel. I truely enjoyed working and discussing with them.

Being a PhD student at KTH has been a wonderful period of my life.

I would like to thank everybody at the department for making it a perfect environment to work in. I thank my fellow PhD students for their good company and friendship.

Last, but not least, it remains to thank my family and friends for their constant support throughout these years.

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Introduction and Summary

1 Schrödinger operators

This thesis is devoted to inequalities for the discrete spectrum of Schrödinger operators

−∆ − V. (1.1)

Here −∆ = − P^dj=1 ∂²

∂x²_j denotes the Laplacian in R^d, d ≥ 1, and V denotes a sufficiently regular, real-valued function. We do not distinguish in the notation between V and the operator of multiplication by V . We shall mostly be interested in the situation where V decays at infinity (at least in some averaged sense) and shall assume this in this introduction. Under broad conditions on V one can consider (1.1) as an unbounded, self-adjoint operator in the Hilbert space L2(R^d) of square-integrable complex-valued functions on R^d.

The spectral theory of Schrödinger operators investigates qualitative and quantitative aspects of the spectrum of the operator (1.1). Generically, this spectrum consists of two parts, namely the continuous spectrum, which cov- ers the whole positive half-line [0, ∞), and the discrete spectrum, which consists of negative eigenvalues of finite multiplicity with 0 as the only possible accumulation point.

Typical questions about the continuous spectrum concern its absolutely or singularly continuous nature, the absence of embedded eigenvalues and the behavior of solutions of the time-dependent Schrödinger equation (scattering theory). In the theory of the discrete spectrum one is interested, for instance, in the behavior of eigenvalues in certain limiting regimes, in uniform bounds in terms of averages of the potential or in properties of the eigenfunctions. Due to its mathematical richness and its physical importance the spectral theory of Schrödinger operators has been developed intensely at

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least since the 1950s. We refer to [BeSh, CyFrKiSi, Da1, Da2, ReSi3, ReSi4]

for textbook presentations of this theory and to the recent review articles [Si2, Si3]. The necessary background in analysis, functional analysis and operator theory is contained, e.g., in [BiSo1, K, LiLo, ReSi1, ReSi2].

We shall not attempt to explain the quantum-mechanical background of the Schrödinger operator (1.1), but this introduction would be incomplete without mentioning the following fundamental concepts. More details may be found e.g. in [Li3, Lo]. In quantum theory a function ψ ∈ L²(R^d) with R |ψ|²dx = 1 is called a wave function and describes the state of a d-dimensional particle (or collection of particles). One interprets |ψ(x)|² as the probability density of finding a particle at x ∈ R^d. The Schrödinger operator (1.1) gives a (simplified) description of the non-relativistic motion of a quantum-mechanical particle in an external potential −V . Its quadratic form

Z

R^d|∇ψ|²dx − Z

R^dV |ψ|²dx

represents the total energy (in suitable units) of the system in the state ψ.

The two terms correspond respectively to the kinetic and the potential part of the total energy. Of particular interest is the ground state energy, i.e. the lowest possible value of the energy, namely

inf

½Z

R^d¡|∇ψ|²− V |ψ|²¢ dx : Z

R^d|ψ|²dx = 1

¾ .

If this infimum is attained by some ψ it coincides with the lowest eigenvalue of the Schrödinger operator −∆ − V . The function ψ is a corresponding eigenfunction and describes the ground state of the system. More generally, the negative eigenvalues of −∆ − V correspond to the possible energy levels and the corresponding eigenfunctions describe bound states of the system.

In the next section we shall introduce the quantities which will be the main objects of our study and recall some known results about them, namely Weyl’s law and the Lieb-Thirring inequalities. In the following Section 3 we shall discuss two mathematical expressions of the uncertainty principle, namely the Sobolev and Hardy inequalities. Sections 4–8 contain summaries of the new results that are contained in this thesis.

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2 The number and moments of negative eigenvalues

The focus of this thesis is on quantitative questions about the discrete spectrum of Schrödinger-type operators. To be more precise, we denote the negative eigenvalues of the operator (1.1) in increasing order by λ₁ ≤ λ2 ≤ . . . ≤ 0, repeating them according to their multiplicities. Their total number is either finite or (if V decays only slowly) countably infinite with zero as the only accumulation point. A central object of study is the total number of negative eigenvalues

N (−∆ − V ) := #{j : λj < 0} (2.1) or, more generally, moments of eigenvalues

tr(−∆ − V )^γ−=X

j

|λj|^γ (2.2)

for some γ > 0. Here and in the sequel x± = (|x| ± x)/2 denote the positive and negative part of numbers, functions and self-adjoint operators. More- over, we shall use the notational convention that tr(−∆−V )⁰−= N (−∆−V ).

In the following subsections we shall discuss two classical results on the number and moments of negative eigenvalues. The first one is Weyl’s law, which describes the asymptotic behavior of the quantities (2.1) and (2.2) in the strong coupling limit, i.e., when V is replaced by αV and α tends to infinity. The second one is a family of inequalities by Cwikel, Lieb and Rozenblyum and by Lieb and Thirring which gives uniform (non-asymptotic) upper bounds for (2.1) and (2.2) in terms of integrals of V .

2.1 Weyl-type asymptotics

A classical result in the spectral theory of Schrödinger operators is Weyl’s law for the number of negative eigenvalues in the strong coupling limit. It states that

α→∞lim α^−d/2N (−∆ − αV ) = (2π)^−dω_d Z

R^d

V (x)^d/2₊ dx, (2.3) where ω_ddenotes the volume of the d-dimensional unit ball. The asymptotics (2.3) were proved by Weyl [We] in 1911 for the Dirichlet Laplacian in a

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bounded domain. Intuitively, this corresponds to a potential V which is constant inside a bounded domain and minus infinity outside of it. (This intuition can be made mathematically precise.) The result for Schrödinger operators appeared for the first time in Birman’s article [Bi] from 1961 for problems in a bounded domain or, which is the same, for V with compact support. In the early seventies it was generalized in [BiBo, Ka, Ma, Ta];

see also [ReSi4, Theorem XIII.80] for a textbook presentation. Initially, the asymptotics (2.3) are established for sufficiently regular V . If d ≥ 3, they are valid for any non-negative V for which the right hand side is finite. The situation in dimensions d = 1 and d = 2 is considerably more subtle. We refer to [BiLa] and the surveys [BiSo2], [BiSo3] concerning this and related issues.

Similarly, for higher moments γ > 0 one has

α→∞lim α^−γ−d/2tr(−∆ − αV )^γ−= L^cl_γ,d Z

R^d

V (x)^γ+d/2₊ dx, (2.4) where

L^cl_γ,d := Γ(γ + 1)

2^dπ^d/2Γ(γ + 1 + d/2).

The asymptotics (2.3) and (2.4) have a semi-classical interpretation. Re- call that the operator (1.1) is the quantization of the Hamiltonian function

|ξ|²− V (x) defined on the classical phase space R^d× R^d. Rewriting (2.4) in terms of a small parameter ~ = α^−1/2 (Planck’s constant) one obtains, in the limit ~ → 0,

tr(−~²∆ − V )^γ−∼ ~^−dL^cl_γ,d Z

R^d

V (x)^γ+d/2₊ dx

= Z Z

R^d×R^d¡|ξ|²− V (x)¢γ

−

dx dξ (2π~)^d.

This is interpreted as each quantum state occupying a volume of (2π~)^d in phase space.

2.2 Lieb-Thirring inequalities

It is a deep fact that the right hand side of (2.4) is not only the limit of the left hand side but actually presents an upper bound, possibly up to a

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multiplicative constant. More precisely, for any γ satisfying γ ≥ 1

2 if d = 1,

γ > 0 if d = 2, (2.5)

γ ≥ 0 if d ≥ 3, there exists a constant L_γ,d such that for any V

tr(−∆ − V )^γ−≤ Lγ,d

Z

R^d

V (x)^γ+d/2₊ dx. (2.6) This inequality was proved by Lieb and Thirring [LiTh2] in the case γ >

max{0, 1 − d/2}. The result in the endpoint case γ = 0, d ≥ 3 is independently due to Cwikel [Cw], Lieb [Li1] and Rozenblyum [Ro1, Ro2] (see also [LYa, Co]) and is usually refered to as the Cwikel-Lieb-Rozenblyum inequality. The endpoint case γ = 1/2, d = 1 was established later by Weidl [Wei]

(see also [HuLiTh]).

We refer to the original papers for the proofs. An essential ingredient in all of them is the Birman-Schwinger principle which we briefly sketch in the following. It transforms the eigenvalue problem of the (unbounded) Schrödinger operator to that of a compact operator. We assume that V is non-negative and sufficiently regular. The basic observation is that if ψ solves (−∆ − V )ψ = −τψ, then ϕ :=√

V ψ solves √

V (−∆ + τ)⁻¹√

V ϕ = ϕ, and conversely, if ϕ solves√

V (−∆ + τ)⁻¹√

V ϕ = ϕ then ψ := (−∆ + τ)⁻¹√ V ϕ solves (−∆−V )ψ = −τψ. Using a monotonicity argument one deduces that the number of eigenvalues less than −τ of the operator −∆−V coincides with the number of eigenvalues bigger than 1 of the operator√

V (−∆ + τ)⁻¹√ V . This principle and a thorough discussion of technical subtleties can be found in Birman’s paper [Bi].

In addition to their connection with Weyl’s asymptotic formula, the Lieb-Thirring inequalities (2.6) are closely related to Sobolev inequalities.

We shall discuss this in detail in the following section. By a non-trivial

‘duality’-argument (2.6) with γ = 1 is equivalent to a lower bound of the quantum-mechanical kinetic energy of fermionic states in terms of a semi- classical kinetic energy. This was one of the essential ingredients in Lieb’s and Thirring’s proof of the stability of matter [LiTh1, Li3]. Inequalities (2.6) have also turned out to be useful in the study of the Navier-Stokes equation and in finding bounds on Riesz and Bessel potentials, see [Li2]. Moreover,

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for such values of γ and d that (2.6) is valid one can use this inequality to extend the Weyl-type asymptotics (2.3), (2.4), which are initially established only for sufficiently regular potentials, to any potential for which the right hand side is finite.

The question about the sharp constants L_γ,d in (2.6) has attracted a lot of attention. So far they are only known for γ ≥ 3/2, d ≥ 1 [LiTh2, AiLi, LaWei1, BeLo], and for γ = 1/2, d = 1 [HuLiTh]; see also the negative result for 0 ≤ γ < 1, d ≥ 1 [HeRo]. In particular, the sharp value of L^1,3 in the physically most important case γ = 1, d = 3 is still unknown. The value has been conjectured [LiTh2] to be L^cl_1,3 from (2.4). We refer to the reviews [BlSt, Li5, LaWei2, Hu] for further information and, in particular, for the connection between Lieb-Thirring inequalities, Berezin-Li-Yau inequalities and the Pólya conjecture.

The conditions (2.5) on γ are optimal. To see this, recall that when d = 1 or d = 2, the Schrödinger operator (1.1) has a negative eigenvalue for any non-negative V 6≡ 0. This follows, for example, from the variational principle by taking the sequence of test functions un(x) := max{1 − |x|/n, 0} if d = 1 and u_n(x) := max{ln(n/|x|)/ ln n, 0}, |x| ≥ 1, and un(x) := 1, |x| < 1, if d = 2. (This shows thatR V dx > 0 is actually sufficient for the existence of a negative eigenvalue.) On the other hand, an inequality of the form (2.6) with γ = 0 implies that there is no negative eigenvalue if R V₊^d/2dx < L⁻¹_0,d. This shows that (2.6) with γ = 0 cannot hold if d = 1 or d = 2.

That (2.6) does not hold for 0 < γ < 1/2, d = 1, can be seen as follows:

Either one chooses a sequence of potentials V which approach a delta function, so that the left hand side of (2.6) has a positive limit while the right hand side tends to zero; or one studies the asymptotic behavior of the lowest eigenvalue λ(α) of the operator −_dx^d²² − αV as α → 0. Using [Si1]

λ(α) ∼ −α µ

1 2

Z

R

V dx

¶2

, α → 0,

if R V dx > 0, one finds again that γ ≥ 1/2 is a necessary condition for (2.6) to hold in one dimension.

We emphasize again that the validity of (2.6) with γ = 0 means that the Schrödinger operator −∆ − V in three or more dimensions does not have negative eigenvalues if the potential V is sufficiently small. Below we shall give a direct proof of this fact using the Hardy and the Sobolev inequalities, which we shall now address.

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3 Hardy and Sobolev inequalities

The inequalities named after Hardy and Sobolev will play a major role in what follows. They are closely related to the uncertainty principle in physics, i.e., a lower bound on the kinetic energy R |∇ψ|²dx in terms of some integral of ψ which does not involve derivatives. Before discussing the Hardy and Sobolev inequalities in detail, we would like to show their use in an example from quantum mechanics.

3.1 The hydrogenic atom

A simplified model of the interaction of an electron with an infinitely heavy nucleus of charge Z > 0 is provided by the Hamiltonian

−∆ − Z|x|⁻¹ in L₂(R³).

Here we have chosen units in which ~²/2m = e = 1. Of course, this is one of the few examples of Schrödinger operators which can be diagonalized explicitly, but here we are looking for more general arguments. Following [Li3] our goal is to understand the reason why this system is stable, i.e., why the electron does not fall into the nucleus leading to an infinite negative energy. That is, we are asking why the ground state energy

E(Z) := inf

½Z

R³

¡|∇ψ|²− Z|x|⁻¹|ψ|²¢ dx : Z

R³|ψ|²dx = 1

¾

(3.1) is finite. Note that the potential energy −Z R |x|⁻¹|ψ(x)|²dx can be made arbitrarily negative by choosing ψ with support in a small neighborhood of the origin (and keeping kψk = 1 fixed). What prevents the collapse is the uncertainty principle: as the wave function becomes more and more localized, its kinetic energy R |∇ψ|²dx has to increase.

Mathematically, this principle is reflected for instance in the Sobolev inequality,

Z

R³|∇ψ|²dx ≥ S³ µZ

R³|ψ|⁶dx

¶1/3

, S3:= 3(π/2)^4/3. (3.2) Combining this inequality with (3.1) and substituting ρ = |ψ|², one obtains

E(Z) ≥ inf (

S₃ µZ

R³

ρ³dx

¶1/3

− Z Z

R³|x|⁻¹ρ dx : Z

R³ρ dx = 1, ρ ≥ 0 )

.

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It is easy to solve the latter minimization problem explicitly. Using the value of S3 in (3.2) one obtains the lower bound

E(Z) ≥ −¹₃Z²,

which is remarkably close to the correct answer, E(Z) = −Z²/4.

A different way to see that E(Z) is finite starts with the Hardy inequality, Z

R³|∇ψ|²dx ≥ ¹₄ Z

R³|x|⁻²|ψ|²dx.

Similarly as above we arrive at an easy minimization problem, E(Z) ≥ inf

½Z

R³

¡₁

4|x|⁻²− Z|x|⁻¹¢ ρ dx : Z

R³ρ dx = 1, ρ ≥ 0

¾ . In this way, we obtain the lower bound E(Z) ≥ −Z².

Having seen this application of the Sobolev and the Hardy inequalities we discuss now several aspects of these inequalities in more detail.

3.2 The Sobolev inequality

A generalization of (3.2) holds in any dimension d ≥ 3 and reads Z

R^d|∇u|²dx ≥ Sd

µZ

R^d|u|^qdx

¶2/q

, q = 2d

d − 2. (3.3) The precise meaning of (3.3) is that if u belongs to the closure of C₀^∞(R^d) with respect to the norm R |∇u|²dx, then u ∈ L^q and (3.3) holds. We refer, e.g., to [LiLo, Chapter 8] for a proof yielding the sharp constant S_dand the fact that there is a unique (up to translation, dilation and multiplication by a constant) function for which equality holds. There one can also find various generalizations, e.g., analogs in dimensions d = 1 and 2 or bounds for L_p-norms with p 6= 2.

There is an important and well-known relation between the Sobolev inequality and (a weak form of) the Cwikel-Lieb-Rozenblyum inequality (2.6) with γ = 0. First note that by duality, (3.3) is equivalent to

Z

R^dV |u|²dx ≤ Z

R^d|∇u|²dx

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for all u and all V ≥ 0 with kV kd/2 ≤ Sd. By the variational principle, this is equivalent to the statement that

−∆ − V ≥ 0 if kV⁺kd/2≤ Sd. (3.4) Hence the Sobolev inequality is equivalent to a condition for the absence of negative eigenvalues of Schrödinger operators.

Several remarks are in order:

(1) In Subsection 2.2 we mentioned that Schrödinger operators in dimension d ≥ 3 do not have negative eigenvalues when the potential is

‘small’. This is quantified by (3.4).

(2) The Cwikel-Lieb-Rozenblyum inequality implies that N(−∆ − V ) = 0 provided kV+k^d/2_d/2 < L⁻¹_0,d. By the equivalence of (3.4) and (3.3) we see that the Cwikel-Lieb-Rozenblyum inequality implies the Sobolev inequality (except possibly for the constant).

(3) The proof of the Cwikel-Lieb-Rozenblyum inequality by Li and Yau [LiYa] relies on the Sobolev inequality. Levin and Solomyak [LeSo]

put this fact into an abstract framework where −∆ is replaced by a much more general operator. Their main result is that, under a certain Markov condition, the Cwikel-Lieb-Rozenblyum inequality is equivalent to the Sobolev inequality.

(4) Lieb’s proof [Li1] of the Cwikel-Lieb-Rozenblyum inequality relies on a pointwise bound on the heat kernel. On the other hand, it is known (see, e.g., [Da1, Chapter 2] and [LiLo, Chapter 8]) that a heat kernel bound is equivalent to a Sobolev inequality. This observation estab- lishes a link between Lieb’s proof and that of Li and Yau. See also [RoSo].

In Paper III we observe that the equivalence to a Sobolev-type inequality does not only hold for the Lieb-Thirring inequality with γ = 0 but for any γ > 0. Below we shall discuss this in some detail.

3.3 The Hardy inequality

Hardy’s original inequality states that for functions u on the halfline R₊ vanishing near 0 and infinity one has

Z ∞

0 |u⁰|²dr ≥ ¹4

Z ∞ 0

r⁻²|u|²dr, (3.5)

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see, e.g., [Da2, Lemma 5.3.1]. (Hardy actually considered an integral version of this inequality.) It is not difficult to deduce from (3.5) that in arbitrary dimension d ≥ 3 one has

Z

R^d|∇u|²dx ≥ ^(d−2)₄ ² Z

R^d|x|⁻²|u|²dx. (3.6) This inequality is initially proved for smooth functions and then extended by continuity to any function in H_loc¹ (R^d) for which the left and the right hand side are finite (the homogeneous Sobolev space). The constants in (3.5) and (3.6) cannot be improved, as may be seen by approximating the function u(x) = |x|^−(d−2)/2 by smooth functions. No minimizer exists, i.e., the inequalities are strict unless u ≡ 0.

The importance of the Hardy inequality for the spectral theory of Schrö- dinger operators was emphasized in Birman’s classical work [Bi]. We note that (3.6) implies that there are no weakly coupled bound states if d ≥ 3.

Indeed,

−∆ − V ≥ 0 if sup |x|²V₊(x) ≤ (d − 2)²/4. (3.7) Note that, in contrast to the integral criterion (3.4), (3.7) requires a pointwise bound on the potential to guarantee the absence of bound states.

There are numerous generalizations of the Hardy inequality (3.6), for example to L_p-norms with p 6= 2 or to domains Ω ⊂ R^d. We refer to [Da3], [OpKu] and the references therein for statements, proofs and applications.

In Paper III we shall recall a version of the Hardy inequality for the pseudo- differential operators (−∆)^s.

4 Overview of paper I. On Lieb-Thirring inequalities for Schrödinger operators with virtual level

Throughout this section we assume that the dimension satisfies d ≥ 3. By the Hardy inequality (3.6) the operator −∆ − ^(d−2)_4|x|²² is non-negative. The question considered in Paper I, joint with T. Ekholm, is whether a Lieb- Thirring-type inequality is valid for the negative eigenvalues of the operator

−∆ −(d − 2)²

4|x|² − V. (4.1)

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Note that a direct approach based only on the inequalities (2.6) and (3.6) would give

tr µ

−∆ − (1 − ε)(d − 2)² 4|x|² − V

¶γ

−≤ tr (−ε∆ − V )^γ−

≤ ε^−d/2L_γ,d Z

R^d

V (x)^γ+d/2₊ dx for any γ ≥ 0 and any ε > 0. However, as ε → 0, the constant on the right hand side diverges.

It is also easy to see that a Cwikel-Lieb-Rozenblyum inequality cannot hold for the operator (4.1). Indeed, for spherically symmetric V the operator (4.1) restricted to spherically symmetric functions coincides with the two-dimensional Schrödinger operator −∆ − V (|x|) restricted to spherically symmetric functions. As we have already mentioned in Subsection 2.2, this operator has a negative eigenvalue for any non-negative V 6≡ 0. A more subtle reason why a Cwikel-Lieb-Rozenblyum inequality cannot hold is the violation of Weyl’s asymptotic law observed by Birman and Laptev [BiLa].

Our main result in Paper I is that a Lieb-Thirring inequality for the operator (4.1) indeed holds as soon as γ > 0. More precisely, we prove Theorem 4.1 (Hardy-Lieb-Thirring inequalities). Let d ≥ 3 and γ >

0. Then tr

µ

−∆ −(d − 2)² 4|x|² − V

¶γ

−

≤ Cγ,d

Z

R^d

V (x)^γ+d/2₊ dx (4.2) with a constant C_γ,d independent of V .

Substituting V −^(d−2)_4|x|²² for V we obtain the inequality

tr (−∆ − V )^γ−≤ Cγ,d

Z

R^d

µ

V (x) −(d − 2)² 4|x|²

¶γ+d/2 +

dx. (4.3) This inequality shows that the part of the potential which is larger than the Hardy weight suffices to control the moments of negative eigenvalues of a Schrödinger operator. In particular, if we replace V by βV , where V is bounded and compactly supported, then the right side of (4.3) is zero for sufficiently small β > 0. This is an important feature of (4.3) which is not shared by the original estimate (2.6).

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Another important aspect of the inequality (4.3) is that it goes beyond semi-classics. If V has a local singularity like c|x|^−α where c > 0, 0 < α ≤ 2, then the semi-classical phase space integral

Z Z

R^d×R^d(|ξ|²− V (x))^γ−

dx dξ (2π)^d

diverges for any sufficiently large γ. Our inequality (4.3) shows that nev- ertheless, tr (−∆ − V )^γ− is finite for any γ > 0 (assuming c < (d − 2)²/4 if α = 2), and can be estimated by an integral of V with the singularity removed.

We mention in passing that the fact that semi-classics underestimates the energy in the presence of singularities is also reflected in the Scott cor- rection for the ground state energy of large atoms or molecules; see [Si2] for references.

The main ingredient in the proof of Theorem 4.1 is an inequality concerning Schrödinger operators on the half-line, which is of independent interest.

We consider the operator −_dr^d²²−_4r¹²−V in L2(R₊) with a Dirichlet boundary condition at the origin, i.e., defined via the closure of the quadratic form

Z

R⁺

µ

|u⁰|²−|u|²

4r² − V |u|²

¶ dr

on C₀^∞(R₊). Recall that 1/4 is the sharp constant in the one-dimensional Hardy inequality (3.5).

Theorem 4.2 (One-dimensional Hardy-Lieb-Thirring inequalities).

Let γ > 0 and α ≥ 1. Then tr

µ

−d² dr² − 1

4r² − V

¶γ

−

≤ Cγ,α

Z

R+

V (r)^γ+

1+α 2

+ r^αdr (4.4)

with a constant Cγ,α independent of V .

Theorem 4.2 is deduced from an inequality of Egorov and Kondrat’ev [EgKo], the proof of which is similar to Rozenblyum’s proof of the Cwikel- Lieb-Rozenblyum inequality.

Several remarks are in order:

(1) The assumption γ > 0 cannot be improved, since the operator −_dr^d²² −

1

4r² has a virtual level, see Proposition 3.2 in Paper I.

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(2) For α = 1 this is a Bargman-type inequality for moments of eigenvalues, see, e.g., [ReSi4, Theorem XIII.9] as well as Remark 1.8 in Paper I.

(3) Paper I contains also a (weak) result in the case 0 ≤ α < 1. This question is completely settled in Paper II.

In Paper III we give an alternative proof of Theorem 4.1 which does not use Theorem 4.2 and which allows for the inclusion of a magnetic field. Before presenting this paper, however, we return to the study of the inequality (4.4).

5 Overview of paper II. Lieb-Thirring inequalities on the half-line with critical exponent

In Paper II, joint with T. Ekholm, we extend the inequality of Theorem 4.2 to weights r^α with 0 ≤ α < 1. Again we consider the Schrödinger operator

−_dr^d²²−_4r¹² − V in L²(R+) with a Dirichlet boundary condition at the origin.

The main result of Paper II is

Theorem 5.1 (One-dimensional Hardy-Lieb-Thirring inequalities).

Let γ > 0 and 0 ≤ α < 1 such that γ +^1+α₂ ≥ 1. Then tr

µ

−d² dr² − 1

4r² − V

¶γ

−≤ Cγ,α

Z

R+

V (r)^γ+

1+α 2

+ r^αdr (5.1)

with a constant C_γ,α independent of V .

The condition γ + ^1+α₂ ≥ 1 in this theorem is optimal, as can be seen by letting V approach a delta function. The important point is that the inequality is valid in the endpoint case γ = ^1−α₂ , when it becomes

tr µ

− d² dr² − 1

4r² − V

¶γ

−≤ C^γ,1−2γ Z

R+

V (r)+r^1−2γdr, 0 < γ ≤ 1

2. (5.2) In particular, this sharpens a result by Egorov and Kondrat’ev [EgKo].

A remarkable implication of (5.2) is that eigenvalue moments of any order 0 < γ ≤ 1/2 can be estimated linearly in V . (For scaling reasons, however, the integral of V now has to include a weight.) This is in contrast to the whole-line case, where only moments of order γ = 1/2 can be estimated linearly, and where moreover the inclusion of a weight does not allow for an extension to smaller values of γ (see [EgKo] and the introduction in Paper II).

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In the important special case α = 0, γ = 1/2 our proof of Theorem 5.1 gives upper and lower bounds on the sharp constant C_1/2,0 in (5.1), which differ by less than a factor 2.25. We compare these bounds with the known sharp constant in (2.6).

Concerning the proof of Theorem 5.1 we first note that by an adaptation of the argument of Aizenman and Lieb [AiLi], the general inequality (5.1) follows from the endpoint inequality (5.2). Our proof of the latter uses ideas from Weidl’s proof of the endpoint Lieb-Thirring inequality [Wei], but its generalization to our non-translation-invariant setting requires additional ideas both on a conceptual and on a technical level. One crucial ingredient in our proof is the combination of Neumann bracketing with the ground-state representation, which will be discussed at the end of the next section.

6 Overview of paper III. Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators

The Lieb-Thirring inequalities for the operator −∆−^(d−2)₄ ²|x|⁻²from Paper I raise the following natural questions:

(1) Does the inequality (4.2) remain true in the presence of a magnetic field, that is, when −∆ is replaced by (−i∇−A)²for a real vector field A?

(2) Does the inequality (4.2) remain true if −∆ is replaced by the pseudo- differential operators (−∆)^s, 0 < s < 1, and |x|⁻² is replaced by

|x|^−2s?

Concerning the first question we note that the Lieb-Thirring inequalities (2.6) remain valid in the presence of a magnetic field with a constant independent of the magnetic field. Indeed, all presently known values of the constant L_γ,d are unchanged when A is included. It is not known whether this is also true for the unknown sharp values. Even though there is a diamagnetic inequality for the lowest eigenvalue, it is known that no such inequality can hold for, e.g., the sums of eigenvalues [Li4], [Ro3].

As for the second question, we first recall that the operator (−∆)^s is defined via the Fourier transform,

((−∆)^su)^∧(ξ) = |ξ|^2su(ξ),ˆ ξ ∈ R^d,

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where we use the convention ˆu(ξ) = (2π)^−d/2R u(x)e^−iξ·xdx. Of particular interest for physics is the case s = 1/2, since √

−∆ represents a pseudo- relativistic kinetic energy and |x|⁻¹ the Coulomb potential.

We shall need the analog of Hardy’s inequality (3.6) for fractional powers of the Laplacian. Since taking roots is operator monotone, (3.6) implies (−∆)^s≥ ((d − 2)/2)^2s|x|^2s for 0 < s < 1. But the constant in this inequality is not sharp, and the condition on s is too restrictive. It was proved independently in [He], [KoPeSe] and [Ya] that the Hardy-type inequality

Z

R^d|ξ|^2s|ˆu(ξ)|²dξ ≥ Cs,d

Z

R^d|x|^−2s|u(x)|²dx, u ∈ C0^∞(R^d), (6.1) holds if and only if 0 < s < d/2, and that the sharp constant is given by

Cs,d:= 2^2sΓ²((d + 2s)/4) Γ²((d − 2s)/4).

In the special case s = 1/2, d = 3 this inequality is also known as Kato’s inequality [K, (V.5.33)].

Given a magnetic vector potential A ∈ L²loc(R^d, R^d) we consider the self- adjoint operator (D − A)² in L²(R^d), where D = −i∇. For 0 < s ≤ 1 we define the operator |D − A|^2s := ((D − A)²)^s by the spectral theorem.

It follows from the diamagnetic inequality (see, e.g., [CyFrKiSi]) that the Hardy inequality (6.1) remains true, with the same constant, if the left hand side is replaced by k|D − A|^suk², 0 < s < 1.

The following theorem proved in Paper III joint with E. H. Lieb and R.

Seiringer, answers the questions posed at the beginning of this section.

Theorem 6.1 (Hardy-Lieb-Thirring inequalities). Let γ > 0, 0 < s ≤ 1 and 0 < s < d/2. Then

tr¡|D − A|^2s− Cs,d|x|^−2s− V¢γ

−≤ Lγ,d,s

Z

R^d

V (x)^γ+d/2s₊ dx (6.2) with a constant L_γ,d,s> 0 independent of V and A.

This result is a significant extension of Theorem 4.1 because the operator (−∆)^s is not a differential operator and it is not ‘local’. Really different techniques will be needed. In particular, we shall use the heat kernel to prove (6.2), in the manner of [Li1]. A bound on this kernel, in turn, will

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be derived from a Sobolev-like inequality which might be of independent interest. Using the non-negative quadratic form

hs[u] :=

Z

R^d|ξ|^2s|ˆu(ξ)|²dξ − Cs,d

Z

R^d|x|^−2s|u(x)|²dx, this bound can be stated as

Theorem 6.2 (Hardy-Sobolev inequality). Let 0 < s ≤ 1, 0 < s < d/2 and 2 ≤ q < 2^∗ = 2d/(d − 2s). Then there exists a constant Cq,d,s > 0 such that

kuk^q≤ Cq,d,shs[u]

d 2s

³₁

2−¹_q´

kuk

d s

³₁

q−_2∗¹´

, u ∈ C0^∞(R^d). (6.3) It is interesting to compare this theorem with the known inequality

kukq≤ Sq,d,sk(−∆)^s/2uk^2s^d

³1 2−¹_q´

kuk^d^s

³1 q−_2∗¹´

for all 2 ≤ q ≤ 2^∗ = 2d/(d − 2s), which follows from the (fractional) Sobolev inequality. Theorem 6.2 says that even when the critical Hardy weight is subtracted, the operator (−∆)^s is still powerful enough to dominate an Lq- norm. However, this q is now required to be strictly less than the critical Sobolev exponent, and one easily checks that the inequality (6.3) fails for q = 2^∗. In the local case, i.e., s = 1, Theorem 6.2 can be derived from an inequality of [BrVa].

As an aside, we note that Theorem 6.2 is one of the main ingredients in our proof of Theorem 6.1. On the other hand, Theorem 6.2 is an easy consequence of Theorem 6.1, except for the values of the constants. This follows by a similar duality argument as in Subsection 3.2.

For proving the equivalence between a Sobolev-type inequality and a heat kernel estimate one needs a contraction property of the heat kernel. In our situation this will be the consequence of the ‘ground state representation formula’. An easy calculation shows that in the sense of distributions one has (−∆)^s|x|^{−(d−2s)/2}= C_s,d|x|^−(d+2s)/2, i.e., |x|^{−(d−2s)/2}can be considered as a

‘generalized ground state’ of the operator (−∆)^s− Cs,d|x|^−2s. Let us recall the ‘ground state representation formula’ in the local case s = 1, already present in [Bi]. If d ≥ 3 and v(x) = |x|^(d−2)/2u(x) then

Z

R^d

µ

|∇u|²−(d − 2)² 4

|u|²

|x|²

¶ dx =

Z

R^d|∇v|² dx

|x|^d−2.

The corresponding formula in the non-local case 0 < s < 1 is more compli- cated but close in spirit.

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Proposition 6.3 (Ground State Representation). Assume that 0 < s <

min{1, d/2}. If u ∈ C0^∞(R^d\ {0}) and v(x) = |x|^(d−2s)/2u(x), then hs[u] = a_s,d

Z

R^d

Z

R^d

|v(x) − v(y)|²

|x − y|^d+2s

dx

|x|^(d−2s)/2 dy

|y|^(d−2s)/2 (6.4) where a_s,d:= 2^2s−1π−d/2 Γ((d+2s)/2)

|Γ(−s)| .

Note that the identity (6.4) gives an independent proof of the Hardy inequality (6.1) for 0 < s < min{1, d/2}. Moreover, it has recently turned out to be useful to obtain remainder estimates in (6.1) [Be].

7 Overview of paper IV. Stability of relativistic matter with magnetic fields for nuclear charges up to the critical value

In Paper IV, joint with E. H. Lieb and R. Seiringer, we present an application and slight improvement of the Hardy-Lieb-Thirring inequalities from Theorem 6.1. We shall give a proof of the ‘stability of relativistic matter’

that goes further than previous proofs by permitting the inclusion of magnetic fields for values of the nuclear charge Z all the way up to the critical value Zα = 2/π.

The term ‘stability of matter’ refers to the observation that the energy content of material objects is (approximately) proportional to the number of particles. While this is an everyday experience (pouring two separate liters of water together produces nothing more exciting than two liters of water), it is a non-trivial fact from a theoretical point of view. Indeed, the number of terms in the expression for the electrostatic energy coming from the electron- electron repulsion and the electron-nucleus attraction grows with the square of the numbers of particles, and not linearly.

Mathematically, the task is to find a lower bound on the ground state energy of the many-body Hamiltonian

H_N,K :=

N

X

j=1

T_j+ αV_N,K(x₁, . . . , x_N; R₁, . . . , R_K) (7.1) in terms of the number of particles. Here N is the number of electrons and the Pauli exclusion principle dictates that H_N,K acts in the anti-symmetric

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N -fold tensor product ∧^NL₂(R³; C^q), where q is the number of spin states of an electron (q = 2 in nature).

In (7.1), T is a (pseudo-)differential operator in L₂(R³) and T_j denotes the operator T acting on the j-th factor in the tensor product ∧^NL₂(R³; C^q).

It represents the kinetic energy of the j-th electron. Several choices of T will be discussed below. The function V_N,K is defined as

V_N,K(x₁, . . . , x_N; R₁, . . . R_K) := X

1≤i<j≤N

|xi− xj|⁻¹

−

N

X

j=1 K

X

k=1

Z_k|xj − Rk|⁻¹+ X

1≤k<l≤K

Z_kZ_l|Rk− Rl|⁻¹.

It describes the potential energy of a system with K fixed nuclei at positions R_k∈ R³ with charges Z_k> 0. The three sums correspond respectively to the electron-electron repulsion, the electron-nucleus attraction and the nucleus- nucleus repulsion. α represents the fine structure constant which in nature equals approximately 1/137.036.

The ground state energy of the quantum-mechanical system described by the Hamiltonian (7.1) is given by inf spec H_N,K. Stability of matter means that the energy per particle is bounded from below (independently of the positions of the nuclei). More precisely, it means that there exists a constant C ≥ 0 such that for all N, K and R1, . . . , R_K one has

inf spec H_N,K ≥ −C(N + K). (7.2)

The existence of such a constant C will, of course, depend on the choice of the kinetic energy T , and sometimes also on the values of the nuclear charges Z_k and the fine structure constant α.

Stability of matter in the non-relativistic case, i.e., T = −(2m)⁻¹∆ in (7.1) where m is the electron mass, was first proved by Dyson and Lenard [DyLe1, DyLe2]. A very elegant proof yielding a constant of the correct order of magnitude was given by Lieb and Thirring [LiTh1]. The Lieb-Thirring inequality (2.6) is a basic ingredient in their proof. We also mention the recent short proof [So]. We refer to [Li3, Li6] and the lecture notes [Lo]

for a thorough discussion of the problem of stability of matter and further references.

To describe relativistic effects one considers the kinetic energy T =

√−∆ + m²− m. Here units are chosen such that the speed of light c = 1.

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The question of stability of matter in the relativistic case is considerably more difficult than in the non-relativistic case and was finally, after important contributions by Conlon, Daubechies, Fefferman and de la Llave, solved by Lieb and Yau in [LiYa]. The important difference to the non-relativistic case is that for (7.2) to hold a bound on α is required in two ways. One is the requirement, for any number of electrons, that Z_jα ≤ 2/π for all j. In fact, if Z_jα > 2/π for some j the Hamiltonian is not bounded below even for a single electron. This is a consequence of the Hardy (or Kato) inequality (6.1). The other requirement is a bound on α itself, α ≤ αc, even for arbitrarily small Z_j > 0, which comes into play when the number of particles is sufficiently large. In [LiYa] the lower bound (7.2) is established for Zjα up to the critical value 2/π and for qα_c≤ 1/47. Since q = 2 and α ≈ 1/137 this is sufficient for physics.

To describe relativistic effects in the presence of a magnetic field one considers the kinetic energy T =p(D − A)²+ m²− m, where A is a magnetic vector potential corresponding to the magnetic field B = curl A. (Note that we absorb the electron charge √α into the vector potential A.) For values of Zjα strictly smaller than the critical value 2/π, it has been shown that stability holds with a magnetic field included, see [LiLoSo] and [LiLoSi].

However, in both proofs the critical value of α goes to zero as Zα approaches 2/π. Paper IV solves this question in the critical case Zα = 2/π.

For the Hamiltonian (7.1) with

T =p(D − A)²+ m²− m and A ∈ L^2,loc(R³, R³), m ≥ 0, we shall prove

Theorem 7.1 (Stability of relativistic matter with magnetic fields).

Let qα ≤ 1/66.5 and αZj ≤ 2/π for all j. Then H_N,K ≥ −mN for all N , K, R₁, . . . , R_K and A.

Note that our upper bound on α in the physical case q = 2 is 1/133, which is worse than the bound from [LiYa] in the non-magnetic case but, fortunately, still larger than the physical value 1/137.

Our proof of Theorem 7.1 follows the strategy of the proof of [LiYa, Theorem 2] but there are two important new ingredients which allow us to include the magnetic field. The first one (Theorem 3.1 in Paper IV) is

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a result on the localization of the magnetic pseudo-relativistic energy. In contrast to the non-relativistic case, the localization error may now depend on the magnetic field. However, we show that it satisfies a diamagnetic inequality. This allows us to carry the non-magnetic estimates of [LiYa] over to our situation. Our second new ingredient (Theorem 4.5 in Paper IV) is an inequality on the sum of eigenvalues of the operator |D − A| − 2/(π|x|) in a ball; we refer to the paper for the definition of the operator |D−A|−2/(π|x|) on a domain. This estimate is a consequence of the Hardy-Lieb-Thirring inequalities from Paper III. In order to improve the constant in this inequality we present another proof for the special case under consideration. It allows us to obtain an eigenvalue estimate in the magnetic case from one in the non-magnetic case at the expense of slightly increasing the constant. In this connection we mention similar results for the number of eigenvalues [Ro3]

and the recent counterexample [FrLoWei] to a magnetic Pólya conjecture.

And now for something completely different...

8 Overview of paper V. Lieb-Thirring inequalities for Schrödinger operators with complex-valued potentials

Throughout the previous sections we assumed the potential V to be a real- valued function, leading to a self-adjoint operator −∆ − V . However, the modeling of certain absorption phenomena in physics gives rise to complex- valued potentials. Recently, these operators have attracted a lot of attention in the mathematical literature, see [Da4] and references therein. Much less is known in the non-self-adjoint case as compared to the self-adjoint case, in particular, since there is no spectral theorem and no variational principle.

If the complex-valued potential V decays sufficiently fast at infinity, the essential spectrum of −∆ − V coincides with [0, ∞) as in the self-adjoint case, but the eigenvalues now can lie anywhere in the complex plane. Again we are interested in estimating them in terms of integrals of the potential.

Apart from their intrinsic interest, such estimates are useful for the numerical computation of eigenvalues.

Paper V, joint with A. Laptev, E. H. Lieb and R. Seiringer, was motivated by a question posed by E. B. Davies. In [AbAsDa] it is proved that when d = 1, any eigenvalue λ ∈ C \ [0, ∞) of the operator (1.1) with arbitrary

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complex-valued integrable potential V satisfies

|λ| ≤ 1 4

µZ

R|V (x)| dx

¶2

. (8.1)

The constant 1/4 in this inequality is sharp as can be seen by letting V approach a delta function. The question was raised whether an analog of (8.1) holds in higher dimensions d ≥ 2 as well.

While we do not answer the question directly, we have succeeded in finding a version of the Lieb-Thirring inequality (2.6) for non-self-adjoint Schrödinger operators. This shows that a bound similar to (8.1) holds in any dimension d ≥ 1 and even for sums of eigenvalues. However, we can only consider eigenvalues outside (arbitrarily small) cones containing the positive real axis and our constant diverges as the angle of this cone tends to zero.

To state our result precisely, we denote by λj, j = 1, 2, 3, . . ., the (countably many) eigenvalues of the Schrödinger operator −∆−V in the cut plane C\[0, ∞), repeated according to their algebraic multiplicities. Recall that the algebraic multiplicity of an eigenvalue λ is the dimension of the generalized eigenspace {ψ : (−∆ − V − λ)^kψ = 0 for some k ∈ N}, which may be larger than the number of linearly independent solutions of (−∆ − V )ψ = λψ, i.e., the geometric multiplicity of λ.

The main result of Paper V is

Theorem 8.1 (Eigenvalue sums). Let d ≥ 1 and γ ≥ 1.

(1) For eigenvalues with non-positive real parts X

<λj<0

(−<λj)^γ≤ Lγ,d

Z

R^d(<V (x))^γ+d/2− dx.

(2) If κ > 0, then for eigenvalues outside the cone {|=z| < κ <z}, X

|=λj|≥κ <λj

|λj|^γ ≤ Cγ,d(κ) Z

R^d|V (x)|^γ+d/2dx.

Here L_γ,d is the same as the sharp constant in (2.6) and C_γ,d(κ) = 2^1+γ/2+d/4¡1 + ²_κ¢γ+d/2

L_γ,d.

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As an easy consequence we obtain Corollary 8.2. Let d ≥ 1 and γ ≥ 1.

(1) For eigenvalues with non-positive real parts X

<λj<0

|λj|^γ≤ Cγ,d

Z

R^d|V (x)|^γ+d/2dx.

(2) If κ > 0, then for eigenvalues inside the cone {|=z| ≤ −κ <z}

X

|=λj|≤−κ <λj

|λj|^γ≤ Lγ,d(κ) Z

R^d(<V (x))^γ+d/2− dx.

Here C_γ,d = 2^1+γ/2+d/4L_γ,d and L_γ,d(κ) = (1 + κ)L_γ,d.

These inequalities are deduced from the Lieb-Thirring inequalites in the self-adjoint case by considering determinantal functions and separating the real and the imaginary part of the quadratic form.

It is natural to conjecture that the estimates in Theorem 8.1 and Corol- lary 8.2 hold for all values of γ for which (2.6) holds, and not only for γ ≥ 1.

If one is interested in single eigenvalues (as in (8.1)), the assumption γ ≥ 1 can be dropped. We refer to Paper V for more details.

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References

[AbAsDa] A. A. Abramov, A. Aslanyan, E. B. Davies, Bounds on complex eigenvalues and resonances, Jour. Phys. A 34, 57-72 (2001).

[AiLi] M. Aizenman, E. H. Lieb, On semiclassical bounds for eigenvalues of Schrödinger operators. Phys. Lett. A 66 (1978), no. 6, 427–429.

[Be] W. Beckner, Pitt’s inequality with sharp error estimates. Pre- print: http://xxx.lanl.gov/math.AP/0701939.

[BeLo] R. D. Benguria, M. Loss, A simple proof of a theorem of Laptev and Weidl. Math. Res. Lett. 7 (2000), 195–203.

[BeSh] F. A. Berezin, M. A. Shubin, The Schrödinger equation. Mathe- matics and its Applications (Soviet Series) 66. Kluwer Academic Publishers Group, Dordrecht, 1991.

[Bi] M. Sh. Birman, The spectrum of singular boundary problems.

Mat. Sb. 55 (1961), 125–174; English transl., Amer. Math. Soc.

Trans. (2) 53 (1966), 23–80.

[BiBo] M. Sh. Birman, V. V. Borzov, On the asymptotics of the discrete spectrum of some singular differential operators. Topics Math.

Phys. 5 (1972), 19–30.

[BiLa] M. Sh. Birman, A. Laptev, The negative discrete spectrum of a two-dimensional Schrödinger operator. Comm. Pure Appl. Math.

49 (1996), 967–997.

[BiSo1] M. Sh. Birman, M. Z. Solomyak, Spectral theory of selfadjoint operators in Hilbert space. Mathematics and its Applications (So- viet Series). D. Reidel Publishing Co., Dordrecht, 1987.

(32)

[BiSo2] M. Sh. Birman, M. Z. Solomyak, Schrödinger operators. Esti- mates for number of bound states as function-theoretic problem.

Amer. Math. Soc. Transl. (2) 150 (1992), 1–54.

[BiSo3] M. Sh. Birman, M. Z. Solomyak, Estimates for the number of negative eigenvalues of the Schrödinger operator and its generalizations. In: Advances in Soviet Mathematics 7, edited by M.

Sh. Birman, 1–55, American Mathematical Society, Providence, RI, 1991.

[BlSt] Ph. Blanchard, J. Stubbe, Bound states for Schrödinger Hamilto- nians: Phase space methods and applications. Rev. Math. Phys.

35(1996), 504–547.

[BrVa] H. Brézis, J.-L. Vázquez, Blow-up solutions of some nonlinear elliptic problems. Rev. Mat. Univ. Comp. Madrid 10 (1997), 443–

469.

[Co] J. Conlon, A new proof of the Cwikel-Lieb-Rosenbljum bound.

Rocky Mountain J. Math. 15 (1985), 117–122.

[CyFrKiSi] H. L. Cycon, R. G. Froese, W. Kirsch, B. Simon, Schrödinger operators with application to quantum mechanics and global geom- etry. Texts and Monographs in Physics. Springer Study Edition.

Springer-Verlag, Berlin, 1987.

[Cw] M. Cwikel, Weak type estimates for singular values and the number of bound states of Schrödinger operators. Ann. Math. 106 (1977), 93–102.

[Da1] E. B. Davies, Heat kernels and spectral theory. Cambridge Tracts in Mathematics 92, Cambridge University Press, Cambridge, 1989.

[Da2] E. B. Davies, Spectral theory and differential operators. Cam- bridge studies in advanced mathematics 42, Cambridge Univer- sity Press, Cambridge, 1990.

[Da3] E. B. Davies, A review of Hardy Inequalities, in: Conference in honour of V. G. Maz’ya, Rostock, 1998. Oper. Theory Adv. Appl.

110, Birkhäuser Verlag, Basel, 1999.

(33)

[Da4] E. B. Davies, Non-self-adjoint differential operators. Bull. Lon- don Math. Soc. 34 (2002), 513–532.

[DyLe1] F. J. Dyson, A. Lenard, Stability of matter. I. J. Math. Phys. 8 (1967), 423–434.

[DyLe2] F. J. Dyson, A. Lenard, Stability of matter. II. J. Math. Phys. 9 (1968), 698–711.

[EgKo] Yu. V. Egorov, V. A. Kondrat’ev, On spectral theory of elliptic operators. Oper. Theory Adv. Appl. 89, Birkhäuser, Basel, 1996.

[FrLoWei] R. L. Frank, M. Loss, T. Weidl, Eigenvalue estimates of the magnetic Laplacian in a domain. Work in progress.

[HeRo] B. Helffer, D. Robert, Riesz means of bounded states and semi- classical limit connected with a Lieb-Thirring conjecture. II. Ann.

Inst. H. Poincaré Phys. Théor. 53 (1990), 139–147.

[He] I. W. Herbst, Spectral theory of the operator (p²+m²)^1/2−Ze²/r.

Comm. Math. Phys. 53 (1977), no. 3, 285–294.

[Hu] D. Hundertmark, Some bound state problems in quantum mechanics. In: Spectral Theory and Mathematical Physics. A Con- ference in Honor of Barry Simon’s 60th Birthday, to appear.

[HuLiTh] D. Hundertmark, E. H. Lieb, L. E. Thomas, A sharp bound for an eigenvalue moment of the one-dimensional Schrödinger operator.

Adv. Theor. Math. Phys. 2 (1998), 719–731.

[Ka] M. Kac, On the asymptotic number of bound states for certain attractive potentials. In: Topics in Functional Analysis, edited by I. Gohberg and M. Kac, 159–167, Academic Press, New York, 1978.

[K] T. Kato, Perturbation theory for linear operators, Reprint of the 1980 edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995.

[KoPeSe] V. Kovalenko, M. Perelmuter and Ya. Semenov, Schrödinger operators with L^l/2w (R^l) potentials. J. Math. Phys. 22 (1981), 1033–

1044.

(34)

[LaWei1] A. Laptev, T. Weidl, Sharp Lieb-Thirring inequalities in high dimensions. Acta Math. 184 (2000), 87–111.

[LaWei2] A. Laptev, T. Weidl, Recent results on Lieb-Thirring inequalities.

Journées “Équations aux Dérivées Partielles" (La Chapelle sur Erdre, 2000), Exp. No. XX, Univ. Nantes, Nantes, 2000.

[LeSo] D. Levin, M. Solomyak, The Rozenblum-Lieb-Cwikel inequality for Markov generators. J. Anal. Math. 71 (1997), 173–193.

[LYa] P. Li, S.-T. Yau, On the Schrödinger equation and the eigenvalue problem. Comm. Math. Phys. 88 (1983), 309–318.

[Li1] E. H. Lieb, The number of bound states of one body Schrödinger operators and the Weyl problem. Proc. A.M.S. Symp. Pure Math.

36(1980), 241–252.

[Li2] E. H. Lieb, Kinetic energy bounds and their application to the stability of matter. In: Schrödinger operators, H. Holden and A.

Jensen eds., Springer Lect. Notes Phys. 345 (1989), 371–382.

[Li3] E. H. Lieb, The stability of matter: From atoms to stars. Bull.

Amer. Math. Soc. 22 (1990), 1–49.

[Li4] E. H. Lieb, Flux phase of the half-filled band. Phys. Rev. Lett.

73(1994), 2158–2161.

[Li5] E. H. Lieb, Lieb-Thirring inequalities. In: Encyclopaedia of Mathematics, Supplement vol. 2, 311-313, Kluwer, 2000.

[Li6] E. H. Lieb, The stability of matter and quantum electrody- namics. In: Proceedings of the Heisenberg symposium, Mu- nich, Dec. 2001, Fundamental physics–Heisenberg and beyond, G. Buschhorn and J. Wess, eds., pp. 53–68, Springer (2004). A modified version appeared in the Milan Journal of Mathematics 71, 199–217 (2003).

[LiLo] E. H. Lieb, M. Loss, Analysis. Second edition. Graduate Studies in Mathematics 14, American Mathematical Society, Providence, RI, 2001.

(35)

[LiLoSi] E. H. Lieb, M. Loss, H. Siedentop, Stability of relativistic matter via Thomas-Fermi theory. Helv. Phys. Acta 69 (1996), 974–984.

[LiLoSo] E. H. Lieb, M. Loss, J. P. Solovej, Stability of matter in magnetic fields. Phys. Rev. Lett. 75 (1995), 985-989.

[LiTh1] E. H. Lieb, W. Thirring, Bound for the kinetic energy of fermi- nons which proves the stability of matter. Phys. Rev. Lett. 35 (1975), 687–689. Errata: ibid., 1116.

[LiTh2] E. H. Lieb, W. Thirring, Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities. Studies in Mathematical Physics, 269–303.

Princeton University Press, Princeton, NJ, 1976.

[LiYa] E. H. Lieb, H.-T. Yau, The stability and instability of relativistic matter. Comm. Math. Phys. 118 (1988), 177–213.

[Lo] M. Loss, Stability of matter. Lecture Notes (2005), http://www.math.gatech.edu/~loss/MUNICH /QUANTUMCOULOMB/coulombwebpage.html

[Ma] A. Martin, Bound states in the strong coupling limit. Helv. Phys.

Acta 45 (1972), 140–148.

[OpKu] B. Opic, A. Kufner, Hardy-type inequalities. Pitman Research Notes in Mathematics Series 219. Longman Scientific & Techni- cal, Harlow, 1990.

[ReSi1] M. Reed, B. Simon, Methods of modern mathematical physics. I.

Functional analysis. Second edition. Academic Press, New York, 1980.

[ReSi2] M. Reed, B. Simon, Methods of modern mathematical physics. II.

Fourier analysis, self-adjointness. Academic Press, New York- London, 1975.

[ReSi3] M. Reed, B. Simon, Methods of modern mathematical physics.

III. Scattering theory. Academic Press, New York-London, 1979.

[ReSi4] M. Reed, B. Simon, Methods of modern mathematical physics.

IV. Analysis of operators. Academic Press, New York-London, 1978.

Hardy-Lieb-Thirring inequalities for eigenvalues of Schrödinger operators